How do i create low lying smoke (similar to Zootopia), in blender
I've tried really simple smoke in Blender and here is what I've got:

Gif with sample

You just need to play with smoke temperature and domain height.
Sphere is nothing more but smoke collision.

UPDATE:

Here it is how to make it:

Gif steps

If you need more info just specify what you want. 

Sorry that I didn't cover timeline for sphere animation.

UPDATE 2

Smoke.


Create a simple plane. Shift+A > Mesh > Plane
With still selected plane go to Physics tab then select Smoke and set smoke type to Flow. Next change the Temp. Diff. to -0.01. It will keep smoke near the ground and even after collision smoke will very slowly descend.





Create cube Shift + A Mesh > Cube and position it as you can see in picture below. It must be a little wider then our plane. So scale it a little S. This cube will be our smoke domain.





Having selected cube go to Physics tab, select Smoke then smoke type Domain. 


First thing you want to change is Resolution, I've set it to 64. Higher values gives more detailes to our smoke.
Secondly you need to check Smoke Adaptive Domain, this will only calculate domain where smoke actually is.
Next check Smoke High Resolution. This will add more details to smoke. Set Resolution to 2 and Noise Method Strenght to 1. Lowering Strenght gives us smooth smoke.






Now save your file and go to Smoke Cache in Domain options. Choose File Format > OpenVDB (this is optional).
Now you can Bake or just hit Alt+A to start animation. Second option will be slow trough te first loop.


As you can see nothing happend which in this case is good :) Now let's make some...

Noise.

We will create collision object and little bit of turbulance to give this smoke a life. 


Create Turbulance - Shift+A > Force field > Turbulance and set Strenght to about 0.5. This option is totally up to you. you can even skip it. What it does is it create some air movements in our domain and based on location (room, outdoor etc.) you will want to adjust position of it and strenght.
Create UV Sphere - Shift+A > Mesh > UV Sphere. In Tools tab select Shading as Smooth. Position sphere in a corner of our domain and right above plane. Check on your timeline if you are on frame 0.





In Top ortho view (sphere selected) hit I and choose Location. Now you can go to frame 50, move your sphere and again hit I and choose Location, and repeat this step. You can also select Automatic keyframe insertion (red dot under the timeline) and do this little faster. See gif below.





With selected sphere go to Physics tab then select Smoke and set smoke type to Collision and Collision type > Animated.


That's it! You can now bake it and play it. Should look like this:



There is still many to do. Smoke texture, lighting, correct density and so on.
If you want I could try to make it but I'm not really good at it.

Here is the blend file:



UPDATE 3



Here I've changed Turbulance to Wind. You can adjust wind speed and noise.
If you want to make it fluffy try higher resolutions. I don't have render farm and 64 is max usable/testing for me.